1.11 Sample test: function i

Chris is 1½ years younger than his brother David. Let $D$ represent David’s age, and $C$ represent Chris’s age.

• A

  If Chris is fifteen years old, how old is David?______

• B

  Write a function to show how to find David’s age, given Chris’s age. $D(C)=$ ______

Sally slips into a broom closet, waves her magic wand, and emerges as…the candy bar fairy! Flying through the window of the classroom, she gives every student two candy bars. Then five candy bars float through the air and land on the teacher’s desk. And, as quickly as she appeared, Sally is gone to do more good in the world.

Let $s$ represent the number of students in the class, and $c$ represent the total number of candy bars distributed. Two for each student, and five for the teacher.

• A

  Write a function to show how many candy bars Sally gave out, as a function of the number of students. $c(s)=$ ______

• B

  Use that function to answer the question: if there were 20 students in the classroom, how many candy bars were distributed? First represent the question in functional notation—then answer it. ______

• C

  Now use the same function to answer the question: if Sally distributed 35 candy bars, how many students were in the class? First represent the question in functional notation—then answer it. ______

The function $f(x)=$ is “Subtract three, then take the square root.”

• A

  Express this function algebraically, instead of in words: $f(x)=$ ______

• B

  Give any three points that could be generated by this function:______

• C

  What $x$ -values are in the domain of this function?______

The function $y\left(x\right)$ is “Given any number, return 6.”

• A

  Express this function algebraically, instead of in words: $y(x)=$ ______

• B

  Give any three points that could be generated by this function:______

• C

  What $x$ -values are in the domain of this function?______

$z(x)=x^2-\mathrm{6x}+9$

• A

  $z(-1)=$ ______

• B

  $z(0)=$ ______

• C

  $z(1)=$ ______

• D

  $z(3)=$ ______

• E

  $z(x+2)=$ ______

• F

  $z(z(x))=$ ______

Of the following sets of numbers, indicate which ones could possibly have been generated by a function. All I need is a “Yes” or “No”—you don’t have to tell me the function! (But go ahead and do, if you want to…)

• A

  $(-\mathrm{2,4})(-\mathrm{1,1})(\mathrm{0,0})(\mathrm{1,1})(\mathrm{2,4})$

• B

  $(\mathrm{4,}-2)(\mathrm{1,}-1)(\mathrm{0,0})(\mathrm{1,1})(\mathrm{4,2})$

• C

  $(\mathrm{2,}\pi )(\mathrm{3,}\pi )(\mathrm{4,}\pi )(\mathrm{5,1})$

• D

  $(\pi ,2)(\pi ,3)(\pi ,4)(\mathrm{1,5})$

Make up a function involving music .

• A

  Write the scenario. Your description should clearly tell me—in words—how one value depends on another.

• B

  Name, and clearly describe, two variables. Indicate which is dependent and which is independent .

• C

  Write a function showing how the dependent variable depends on the independent variable. If you were explicit enough in parts (a) and (b), I should be able to predict your answer to part (c) before I read it .

• D

  Choose a sample number to show how your function works. Explain what the result means.

Here is an algebraic generalization: for any number $x$ , $x^2-\text{25}=(x+5)(x-5)$ .

• A

  Plug $x=3$ into that generalization, and see if it works.

• B

  $\text{20}\times \text{20}$ is 400. Given that, and the generalization, can you find $\text{15}\times \text{25}$ without a calculator? (Don’t just give me the answer, show how you got it!)

Amy has started a company selling candy bars. Each day, she buys candy bars from the corner store and sells them to students during lunch. The following graph shows her profit each day in March.

• A

  On what days did she break even?

• B

  On what days did she lose money?

The picture below shows the graph of $y=\sqrt{x}$ . The graph starts at $(\mathrm{0,0})$ and moves up and to the right forever.

• A

  What is the domain of this graph?

• B

  Write a function that looks exactly the same, except that it starts at the point $(-\mathrm{3,1})$ and moves up-and-right from there.

The following graph represents the graph $y=f(x)$ .

• A

  Is it a function? Why or why not?

• B

  What are the zeros?

• C

  For what $x-\mathrm{values}$ is it positive?

• D

  For what $x-\mathrm{values}$ is it negative?

• E

  Below is the same function $f(x)$ . On that same graph, draw the graph of $y=f(x)-2$ .

• F

  Below is the same function $f(x)$ . On that same graph, draw the graph of $y=-f(x)$ .